Question

Find the derivative using the first principle: sin e^x

Answer 1

Answer:

To find the derivative of sinx, we return to the first principles definition of the derivative of y=f(x): dydx=limh→0f(x+h)−f(x)h. dydx=limh→0sin(x+h)−sinxh.

Step-by-step explanation:

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Answer 2

Answer:

Step-by-step explanation:

Concept:

The Leibniz notation can also be used to represent the chain rule. If two variables $y$ and $z$ are dependent variables (i.e., $y$ and $z$ are dependent variables), then $z$ is also reliant on $x$ through the intermediary variable $y$. The chain rule is written as follows in this scenario:

$\frac{dz}{dx} =\frac{dz}{dy} .\frac{dy}{dx}$

For composites made up of more than two functions, the chain rule can be used. Remember that the composite of $f, g,$ and $h$ (in that sequence) is the composite of $f$ with $g$ ∘ $h$  in order to calculate the derivative of a composite of more than two functions. According to the chain rule, it is sufficient to compute the derivative of $f$ and the derivative of $g$ ∘ $h$  in order to calculate the derivative of $f$ ∘ $g$ ∘ $h$. You can determine the derivative of $f$ directly, and you can determine the derivative of $g$ ∘ $h$ by reiterating the chain rule.

Given:

The derivative $sin e^x$

Find:

We have to find the derivative using the first principle $sin e^x$

Solution:

Given that

The derivative $sin e^x$

Using the chain rule along with the known derivatives

$\frac{d}{dx} sinx=cosx\\\frac{d}{dx}e^x=e^x$

We have

$\frac{dy}{dx} =\frac{d}{dx} sine^x\\=cose^x(\frac{d}{dx} e^x)\\=cose^x(cose^x)\\=e^x(cose^x)$

$\frac{d}{dx} sine^x=e^x cos(e^x)$

Hence $\frac{d}{dx} sine^x=e^xcos(e^x)$

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